Understanding Uncountable Well-Ordered Sets: An Intuitive Explanation

[zefram_c]

I can't get a good intuitive grasp on this set. Folland defines it as follows:

There is an uncountable well ordered set S such that I_x={y in S: y<x} is countable for each x in S. If S' is another set with the same properties, then S and S' are order isomorphic.

Proof: Uncountable well-ordered sets exist by the WEP, let X be one. Either X has the desired property or there is a minimal element x₀ s.t. I_x0 is uncountable, in which case let S be I_x0

My questions / problems with it are somewhat as follows:

1) I don't see how the set can be uncountable if any initial segment is countable.

2) How do we know that the element x₀ used in the proof exists?

 

[Hurkyl]

If you want an analogy, consider the first infinite ordinal, $\omega$. While it is an infinite ordinal, but every initial segment is finite... if you can understand this, you should be able to understand the first uncountable ordinal.


x₀ exists because ordinals are well ordered. Any collection of ordinals has a smallest element.

 

[mathwonk]

Did you understand Hurkyl's response to your question?

I.e. either you meant to ask: why does it follow logically that the statements given are true? (which is an immediate consequence of their definitions and of logic)

or you meant to ask: "I understand the proof, but how can this be possible?"

Hurkyl answered the second version of your question.

re-reading, it seems your question 1) was the psychological one,
and your 2) was a (tauto)logical one.

ah yes, you said you wanted an intuitive grasp of the set, so Hurkyl understood you correctly.

 

[zefram_c]

Thanks Hurkyl - after reading your response, I went back and checked. Turns out my understanding of well-ordering was incorrect; I now get the logic. The analogy gave me food for thought, so I will think on it for now and see if I can persuade myself that it works.